Optimization problems arising in engineering design often exhibit specific features which, in the interest of computational efficiency, ought to be exploited. Such is the possible presence of 'functional' specifications, i.e., specifications that are to be met over an interval of values of an independent parameter such as time or frequency. While problems involving such specifications could be handled by general purpose nondifferentiable optimization algorithms, the particular structure of functional constraints calls for specific techniques. Suitable schemes have been proposed in the literature. Global convergence is typically achieved by making use of some kind of adaptively refined discretization of the interval of variation of the independent parameter. One previously proposed algorithm exploits the regularity properties of the functions involved to dramatically-reduce the computational overhead incurred once the discretization mesh becomes small. In this paper examples are given that show however that, if the initial discretization is coarse, convergence to a nonstationary point may occur. The cause of such failure is investigated and a class of algorithms is proposed that circumvent this difficulty.